When annuities do not commence till a certain period of Time has elapsed, or till some particular event has taken place, they are said to be in reversion.

9. What is the present worth of $100 annuity, to be continued 4 years, but not to commence till 2 years hence, allowing 6 per cent. compound interest?

The present worth is evidently a sum which, at 6 per cent. compound interest, would in 2 years produce an amount equal to the present worth of the annuity, were it to commence immediately. By the last rule, we find the present worth of the annuity, to commence immediately, to be $346'51, and, by directions under ¶ 114, ex. 4, we find the present worth of $346'51 for 2 years, to be $308'393. Ans. $308'393.

Hence, to find the present worth of any annuity taken in reversion, at compound interest,-First, find the present worth, to commence immediately, and this sum, divided by the power of the ratio, denoted by the time in reversion, will give the

answer.

10. What ready money will purchase the reversion of a lease of $60 per annum, to continue 6 years, but not to commence till the end of 3 years, allowing 6 per cent. compound interest to the purchaser ?

The present worth, to commence immediately, we find to Ans. $247'72.

be, $295'039, and

295'039
1'063

=247'72.

It is plain, the same result will be obtained by finding the present worth of the annuity, to commence immediately, and to continue to the end of the time, that is, 3+6=9 years, and then subtracting from this sum the present worth of the annuity, continuing for the time of reversion, 3 years. Or, we may find the present worth of $1 for the two times by the table, and multiply their difference by the given annuity. Thus, by the table,

The whole time,

9 years, 6'80169 The time in reversion, 3 years, = 2'67301

Difference, =4'12868

60

$247'72080 Ans.

11. What is the present worth of a lease of $100 to continue 20 years, but not to commence till the end of 4 years,

what, if it be 6 years in rever14 years?

allowing 5 per cent. ? 8 years?

sion?

10 years?

Ans. to last, $629'426.

¶ 117. 12. What is the worth of a freehold estate, of which the yearly rent is $60, allowing to the purchaser 6 per cent.?

In this case, the annuity continues forever, and the estate is evidently worth a sum, of which the yearly interest is equal to the yearly rent of the estate. The principal multiplied by the rate gives the interest; therefore, the interest divided by the rate will give the principal; 60÷'06 = 1000.

Ans. $1000.

Hence, to find the present worth of an annuity, continuing forever,-Divide the annuity by the rate per cent., and the quotient will be the present worth.

Note. The worth will be the same, whether we reckon simple or compound interest; for, since a year's interest of the price is the annuity, the profits arising from that price can neither be more nor less than the profits arising from the annuity, whether they be employed at simple or compound in

terest.

13. What is the worth of $100 annuity, to continue forever, allowing to the purchaser 4 per cent.?

5 per cent.?

per cent. ?

8 per cent.? 20 per cent.?

allowing

15

10 per cent.?
Ans. to last, $500.

14. Suppose a freehold estate of $60 per annum, to commence 2 years hence, be put on sale; what is its value, allowing the purchaser 6 per cent. ?

Its present worth is a sum which, at 6 per cent. compound interest, would, in 2 years, produce an amount equal to the worth of the estate if entered on immediately.

60 $1000 the worth, if entered on immediately,

406

and

$1000

1'062

The same result may be obtained by subtracting from the worth of the estate, to commence immediately, the present worth of the annuity 60, for 2 years, the time of REVERSION. Thus, by the table, the present worth of $1 for 2 years is 1'83339 X 60 110'0034 present worth of $60 for 2 years, and $1000 $110'0034 $889'9966, Ans. as before.

15. What is the present worth of a perpetual annuity of $100, to commence 6 years hence, allowing the purchaser 5 per cent. compound interest? what, if 8 years in reversion? ——— 10 years ? ——— 4 years? -15 years? 30 years? Ans. to last, $462'755.

The foregoing examples, in compound interest, have been confined to yearly payments; if the payments are half yearly, we may take half the principal or annuity, half the rate per cent., and twice the number of years, and work as before, and so for any other part of a year.

QUESTIONS.

the

1. What is a geometrical progression or series? 2. What is the ratio? 3. When the first term, the ratio, and the number of terms, are given, how do you find the last term? 4. When the extremes and ratio are given, how do you find the sum of all the terms? 5. When the first term, the ratio, and the number of terms, are given, how do you find the amount of the series? 6. When the ratio is a fraction, how do you proceed? 7. What is compound interest? 8. How does it appear that the amounts, arising by compound interest, form a geometrical series? 9. What is the ratio, in compound interest? the number of terms? first term? the lust term? 10. When the rate, the time, and the principal, are given, how do you find the amount? 11. When A. R. and T. are given, how do you find P.? 12. When A. P. and T. are given, how do you find R.? 13. When A. P. and R. are given, how do you find T.? 14. What is an annuity? 15. When are annuities said to be in arrears? 16. What is the amount? 17. In a geometrical series, to what is the amount of an annuity equivalent? 18. How do you find the amount of an annuity, at compound interest? 19. What is the present worth of an annuity? how computed at compound interest? how found by the table? 20. What is understood by the term reversion? 21. How do you find the present worth of an annuity, taken in reversion? by the table? 22. How do you find the worth of a freehold estate, or a perpetual annuity? same taken in reversion? by the table?

present the

PERMUTATION.

¶ 118. Permutation is the method of finding how many different ways the order of any number of things may be varied or changed.

1. Four gentlemen agreed to dine together so long as they could sit, every day, in a different order or position; how many days did they dine together?

Had there been but two of them, a and b, they could sit only in 2 times 1 (1 X 2 = 2) different positions, thus, a b, and b a. Had there been three, a, b, and c, they could sit in 1 X 2 X 36 different positions; for, beginning the order with a, there will be 2 positions, viz. a b c, and a cb; next, beginning with b, there will be 2 positions, ba c, and bca; lastly, beginning with c, we have c a b, and c b a, that is, in all, 1 X 2 × 3 = 6 different positions. In the same manner, if there be four, the different positions will be 1 × 2 × 3 × 4 = 24. Ans. 24 days.

Hence, to find the number of different changes or permutations, of which any number of different things are capable,Multiply continually together all the terms of the natural series of numbers, from 1 up to the given number, and the last product will be the answer.

2. How many variations may there be in the position of the nine digits? Ans. 362880. 3. A man bought 25 cows, agreeing to pay for them 1 cent for every different order in which they could all be placed; how much did the cows cost him?

Ans. $155112100433309859840000. 4. Christ Church, in Boston, has 8 bells; how many changes may be rung upon them?

Ans. 40320.

MISCELLANEOUS EXAMPLES.

¶ 119. 1. 4+6 × 7—1=60.

A line, or vinculum, drawn over several numbers, signifies, that the numbers under it are to be taken jointly, or as one whole number.

2.9-84X8+4-6-how many?

3.7+4-2+ 3+ 40 x 5 how many?

=

Ans. 30.

Ans. 230.

Ans. 34.

5. There are two numbers; the greater is 25 times 78, and their difference is 9 times 15; their sum and product are required.

Ans. 3765 is their sum; 3539250 their product. 6. What is the difference between thrice five and thirty, and thrice thirty-five? 35 X 3-5 X 3 +30 60, Ans. 7. What is the difference between six dozen dozen, and half a dozen dozen?

Ans. 792.

8. What number divided by 7 will make 6488?
9. What number multiplied by 6 will make 2058?

10. A gentleman went to sea at 17 years of age; 8 years after he had a son born, who died at the age of 35; after whom the father lived twice 20 years; how old was the father at his death? Ans. 100 years.

the product will be ?

11. What number is that, which being multiplied by 15, 215, Ans. 12. What decimal is that, which being multiplied by 15, the product will be '75 ? 675 ÷ 15—‘05, Ans.

13. What is the decimal equivalent to?

Ans. '0285714. 14. What fraction is that, to which if you add, the sum will be ? 15. What number is that, from which if you take 3, the remainder will be ?

Ans. 18

Ans. 28

16. What number is that, which being divided by, the quotient will be 21 ?

Ans. 15.

17. What number is that, which multiplied by

"

duces ?

18. What number is that, from which if you itself, the remainder will be 12? 19. What number is that, to which if you add

itself, the whole will be 20?

20. What number is that, of which 9 is the

proAns. .

part?

Ans. 134. market: he

21. A farmer carried a load of produce to sold 780 lbs. of pork, at 6 cents per lb.; 250 lbs. of cheese, at 8 cents per lb.; 154 lbs. of butter, at 15 cents per lb.: